Theorem tz7.48-3 8445

Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

Hypothesis

Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.48-3	⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-3

Step	Hyp	Ref	Expression
1		tz7.48.1	. . . . 5 ⊢ 𝐹 Fn On
2	1	fndmi 6647	. . . 4 ⊢ dom 𝐹 = On
3		onprc 7762	. . . 4 ⊢ ¬ On ∈ V
4	2, 3	eqneltri 2846	. . 3 ⊢ ¬ dom 𝐹 ∈ V
5	1	tz7.48-2 8443	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)
6		funrnex 7939	. . . . . 6 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V))
7	6	com12 32	. . . . 5 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V))
8		df-rn 5680	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
9	8	eleq1i 2818	. . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
10		dfdm4 5889	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
11	10	eleq1i 2818	. . . . 5 ⊢ (dom 𝐹 ∈ V ↔ ran ◡𝐹 ∈ V)
12	7, 9, 11	3imtr4g 296	. . . 4 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
13	5, 12	syl 17	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
14	4, 13	mtoi 198	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V)
15	1	tz7.48-1 8444	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)
16		ssexg 5316	. . . 4 ⊢ ((ran 𝐹 ⊆ 𝐴 ∧ 𝐴 ∈ V) → ran 𝐹 ∈ V)
17	16	ex 412	. . 3 ⊢ (ran 𝐹 ⊆ 𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V))
18	15, 17	syl 17	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V))
19	14, 18	mtod 197	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098  ∀wral 3055  Vcvv 3468   ∖ cdif 3940   ⊆ wss 3943  ^◡ccnv 5668  dom cdm 5669  ran crn 5670   “ cima 5672  Oncon0 6358  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  tz7.49  8446