Theorem fompt 39379

Description: Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
fompt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
fompt	⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑦,𝐹

Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fompt

Step	Hyp	Ref	Expression
1		fompt.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		nfmpt1 4747	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3	1, 2	nfcxfr 2762	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	dffo3f 39364	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
5	4	simplbi 476	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	1	fmpt 6381	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
7	6	bicomi 214	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
8	7	biimpi 206	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
9	5, 8	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
10	3	foelrnf 39373	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
11		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
13	3, 11, 12	nffo 6114	. . . . . . 7 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵
14		simpr 477	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
15		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
16	9	r19.21bi 2932	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
17	1	fvmpt2 6291	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
18	15, 16, 17	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
19	18	adantr 481	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) = 𝐶)
20	14, 19	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = 𝐶)
21	20	ex 450	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶))
22	21	ex 450	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶)))
23	13, 22	reximdai 3012	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
24	23	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
25	10, 24	mpd 15	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
26	25	ralrimiva 2966	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
27	9, 26	jca 554	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
28	6	biimpi 206	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
29	28	adantr 481	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴⟶𝐵)
30		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
31		nfra1 2941	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶
32	30, 31	nfan 1828	. . . . 5 ⊢ Ⅎ𝑦(∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
33		simpll 790	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
34		rspa 2930	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
35	34	adantll 750	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
36		nfra1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
37		simp3 1063	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = 𝐶)
38		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
39		rspa 2930	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
40	38, 39, 17	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
41	40	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐹‘𝑥))
42	41	3adant3 1081	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = (𝐹‘𝑥))
43	37, 42	eqtrd 2656	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = (𝐹‘𝑥))
44	43	3exp 1264	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → 𝑦 = (𝐹‘𝑥))))
45	36, 44	reximdai 3012	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
46	45	imp 445	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
47	33, 35, 46	syl2anc 693	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
48	47	ex 450	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
49	32, 48	ralrimi 2957	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
50	29, 49	jca 554	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
51	50, 4	sylibr 224	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴–onto→𝐵)
52	27, 51	impbii 199	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ↦ cmpt 4729  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  disjinfi  39380