Theorem hashf1lem1 13239

Description: Lemma for hashf1 13241. (Contributed by Mario Carneiro, 17-Apr-2015.)

Hypotheses

Ref	Expression
hashf1lem2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashf1lem2.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashf1lem2.3	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashf1lem2.4	⊢ (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))
hashf1lem1.5	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

Assertion

Ref	Expression
hashf1lem1	⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Distinct variable groups:   𝑧,𝑓   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓   𝑓,𝐹

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝐹(𝑧)

Proof of Theorem hashf1lem1

Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6101	. . . . . 6 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
2	1	adantl 482	. . . . 5 ⊢ (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3		hashf1lem2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin)
4		hashf1lem2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		snfi 8038	. . . . . . 7 ⊢ {𝑧} ∈ Fin
6		unfi 8227	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
7	4, 5, 6	sylancl 694	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
8	3, 7	elmapd 7871	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
9	2, 8	syl5ibr 236	. . . 4 ⊢ (𝜑 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧}))))
10	9	abssdv 3676	. . 3 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})))
11		ovex 6678	. . 3 ⊢ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ∈ V
12		ssexg 4804	. . 3 ⊢ (({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ∧ (𝐵 ↑𝑚 (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
13	10, 11, 12	sylancl 694	. 2 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
14		difexg 4808	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
15	3, 14	syl 17	. 2 ⊢ (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16		vex 3203	. . . 4 ⊢ 𝑔 ∈ V
17		reseq1 5390	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝐴) = (𝑔 ↾ 𝐴))
18	17	eqeq1d 2624	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ↾ 𝐴) = 𝐹 ↔ (𝑔 ↾ 𝐴) = 𝐹))
19		f1eq1 6096	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
20	18, 19	anbi12d 747	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)))
21	16, 20	elab 3350	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
22		f1f 6101	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
23	22	ad2antll 765	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24		ssun2 3777	. . . . . . 7 ⊢ {𝑧} ⊆ (𝐴 ∪ {𝑧})
25		vex 3203	. . . . . . . 8 ⊢ 𝑧 ∈ V
26	25	snss 4316	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
27	24, 26	mpbir 221	. . . . . 6 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
28		ffvelrn 6357	. . . . . 6 ⊢ ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔‘𝑧) ∈ 𝐵)
29	23, 27, 28	sylancl 694	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
30		hashf1lem2.3	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
31	30	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ 𝑧 ∈ 𝐴)
32		df-ima 5127	. . . . . . . . 9 ⊢ (𝑔 “ 𝐴) = ran (𝑔 ↾ 𝐴)
33		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 ↾ 𝐴) = 𝐹)
34	33	rneqd 5353	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ran (𝑔 ↾ 𝐴) = ran 𝐹)
35	32, 34	syl5eq 2668	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 “ 𝐴) = ran 𝐹)
36	35	eleq2d 2687	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ (𝑔‘𝑧) ∈ ran 𝐹))
37		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
38	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39		ssun1 3776	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
40	39	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41		f1elima 6520	. . . . . . . 8 ⊢ ((𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
42	37, 38, 40, 41	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
43	36, 42	bitr3d 270	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐴))
44	31, 43	mtbird 315	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ (𝑔‘𝑧) ∈ ran 𝐹)
45	29, 44	eldifd 3585	. . . 4 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹))
46	45	ex 450	. . 3 ⊢ (𝜑 → (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
47	21, 46	syl5bi 232	. 2 ⊢ (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48		hashf1lem1.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
49		f1f 6101	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
50	48, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
51	50	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴⟶𝐵)
52		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	25, 52	f1osn 6176	. . . . . . 7 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54		f1of 6137	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
55	53, 54	ax-mp 5	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56		eldifi 3732	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥 ∈ 𝐵)
57	56	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥 ∈ 𝐵)
58	57	snssd 4340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59		fss 6056	. . . . . 6 ⊢ (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
60	55, 58, 59	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61		res0 5400	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
62		res0 5400	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
63	61, 62	eqtr4i 2647	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64		disjsn 4246	. . . . . . . . 9 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
65	30, 64	sylibr 224	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
66	65	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
67	66	reseq2d 5396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
68	66	reseq2d 5396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
69	63, 67, 68	3eqtr4a 2682	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70		fresaunres1 6077	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
71	51, 60, 69, 70	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72		f1f1orn 6148	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
73	48, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
74	73	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
75	53	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76		eldifn 3733	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
77	76	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78		disjsn 4246	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
79	77, 78	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80		f1oun 6156	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→ran 𝐹 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}) ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ (ran 𝐹 ∩ {𝑥}) = ∅)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
81	74, 75, 66, 79, 80	syl22anc 1327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82		f1of1 6136	. . . . . 6 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
84		frn 6053	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
85	51, 84	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹 ⊆ 𝐵)
86	85, 58	unssd 3789	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
87		f1ss 6106	. . . . 5 ⊢ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
88	83, 86, 87	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
89		fex 6490	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ V)
90	50, 4, 89	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
91	90	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
92		snex 4908	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩} ∈ V
93		unexg 6959	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
94	91, 92, 93	sylancl 694	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
95		reseq1 5390	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓 ↾ 𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
96	95	eqeq1d 2624	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓 ↾ 𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
97		f1eq1 6096	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵))
98	96, 97	anbi12d 747	. . . . . 6 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
99	98	elabg 3351	. . . . 5 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
100	94, 99	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
101	71, 88, 100	mpbir2and 957	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
102	101	ex 450	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
103	21	anbi1i 731	. . 3 ⊢ ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
104		simprlr 803	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
105		f1fn 6102	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔 Fn (𝐴 ∪ {𝑧}))
106	104, 105	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
107	81	adantrl 752	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
108		f1ofn 6138	. . . . . . 7 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
109	107, 108	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
110		eqfnfv 6311	. . . . . 6 ⊢ ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
111	106, 109, 110	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
112		fvres 6207	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝑔 ↾ 𝐴)‘𝑦) = (𝑔‘𝑦))
113	112	eqcomd 2628	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) = ((𝑔 ↾ 𝐴)‘𝑦))
114		simprll 802	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 ↾ 𝐴) = 𝐹)
115	114	fveq1d 6193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
116	113, 115	sylan9eqr 2678	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = (𝐹‘𝑦))
117	48	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
118		f1fn 6102	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
119	117, 118	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
120	25, 52	fnsn 5946	. . . . . . . . . . 11 ⊢ {⟨𝑧, 𝑥⟩} Fn {𝑧}
121	120	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
122	65	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝐴 ∩ {𝑧}) = ∅)
123		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
124		fvun1 6269	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ {⟨𝑧, 𝑥⟩} Fn {𝑧} ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ 𝑦 ∈ 𝐴)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
125	119, 121, 122, 123, 124	syl112anc 1330	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
126	116, 125	eqtr4d 2659	. . . . . . . 8 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
127	126	ralrimiva 2966	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
128	127	biantrurd 529	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
129		ralunb 3794	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
130	128, 129	syl6bbr 278	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
131	25	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑧 ∈ V)
132	52	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑥 ∈ V)
133		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
134	50, 133	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
135	134	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
136	30, 135	mtbird 315	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
137	136	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
138		fsnunfv 6453	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
139	131, 132, 137, 138	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
140	139	eqeq2d 2632	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔‘𝑧) = 𝑥))
141		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑔‘𝑦) = (𝑔‘𝑧))
142		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
143	141, 142	eqeq12d 2637	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
144	25, 143	ralsn 4222	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
145		eqcom 2629	. . . . . 6 ⊢ (𝑥 = (𝑔‘𝑧) ↔ (𝑔‘𝑧) = 𝑥)
146	140, 144, 145	3bitr4g 303	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔‘𝑧)))
147	111, 130, 146	3bitr2d 296	. . . 4 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧)))
148	147	ex 450	. . 3 ⊢ (𝜑 → ((((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
149	103, 148	syl5bi 232	. 2 ⊢ (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
150	13, 15, 47, 102, 149	en3d 7992	1 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955  1c1 9937   + caddc 9939   ≤ cle 10075  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959

This theorem is referenced by:  hashf1lem2  13240