Theorem tz7.48-3 8463

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 7777	. . . 4 ⊢ ¬ On ∈ V
4	2, 3	eqneltri 2854	. . 3 ⊢ ¬ dom 𝐹 ∈ V
5	1	tz7.48-2 8461	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)
6		funrnex 7957	. . . . . 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 5670	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
9	8	eleq1i 2826	. . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
10		dfdm4 5880	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
11	10	eleq1i 2826	. . . . 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 199	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V)
15	1	tz7.48-1 8462	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)
16		ssexg 5298	. . . 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 198	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∖ cdif 3928   ⊆ wss 3931  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662  Oncon0 6357  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  tz7.49  8464