Theorem tz7.49c 8247

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 260	. . 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 8246	. 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 1148	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
5		onss 7611	. . . . . . . . 9 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
6		fnssres 6539	. . . . . . . . 9 ⊢ ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹 ↾ 𝑥) Fn 𝑥)
7	1, 5, 6	sylancr 586	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐹 ↾ 𝑥) Fn 𝑥)
8		df-ima 5593	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
9	8	eqeq1i 2743	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ran (𝐹 ↾ 𝑥) = 𝐴)
10	9	biimpi 215	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) = 𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
11	7, 10	anim12i 612	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
12	11	anim1i 614	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
13		dff1o2 6705	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
14		3anan32 1095	. . . . . . 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 274	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
16	12, 15	sylibr 233	. . . . 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 3172	. 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 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582   “ cima 5583  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  –1-1-onto→wf1o 6417  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  dfac8alem  9716  dnnumch1  40785