Theorem tz7.49c 7541

Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)

Hypothesis

Ref	Expression
tz7.49c.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.49c	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem tz7.49c

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tz7.49c.1	. . 3 ⊢ 𝐹 Fn On
2		biid 251	. . 3 ⊢ (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
3	1, 2	tz7.49 7540	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
4		3simpc 1060	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
5		onss 6990	. . . . . . . . 9 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
6		fnssres 6004	. . . . . . . . 9 ⊢ ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹 ↾ 𝑥) Fn 𝑥)
7	1, 5, 6	sylancr 695	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐹 ↾ 𝑥) Fn 𝑥)
8		df-ima 5127	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
9	8	eqeq1i 2627	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ran (𝐹 ↾ 𝑥) = 𝐴)
10	9	biimpi 206	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) = 𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
11	7, 10	anim12i 590	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
12	11	anim1i 592	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
13		dff1o2 6142	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
14		3anan32 1050	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
15	13, 14	bitri 264	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
16	12, 15	sylibr 224	. . . . 5 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
17	16	expl 648	. . . 4 ⊢ (𝑥 ∈ On → (((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴))
18	4, 17	syl5 34	. . 3 ⊢ (𝑥 ∈ On → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴))
19	18	reximia 3009	. 2 ⊢ (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
20	3, 19	syl 17	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘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-rep 4771  ax-sep 4781  ax-nul 4789  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-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-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  dfac8alem  8852  dnnumch1  37614