Theorem tz7.49c 8417

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 261	. . 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 8416	. 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 1150	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
5		onss 7764	. . . . . . . . 9 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
6		fnssres 6644	. . . . . . . . 9 ⊢ ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹 ↾ 𝑥) Fn 𝑥)
7	1, 5, 6	sylancr 587	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐹 ↾ 𝑥) Fn 𝑥)
8		df-ima 5654	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
9	8	eqeq1i 2735	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ran (𝐹 ↾ 𝑥) = 𝐴)
10	9	biimpi 216	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) = 𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
11	7, 10	anim12i 613	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
12	11	anim1i 615	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
13		dff1o2 6808	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
14		3anan32 1096	. . . . . . 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 275	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
16	12, 15	sylibr 234	. . . . 5 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
17	16	expl 457	. . . 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 3065	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299  ^◡ccnv 5640  ran crn 5642   ↾ cres 5643   “ cima 5644  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  dfac8alem  9989  dnnumch1  43040