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Theorem tz7.48-3 7404
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 (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-3
StepHypRef Expression
1 onprc 6854 . . . 4 ¬ On ∈ V
2 tz7.48.1 . . . . . 6 𝐹 Fn On
3 fndm 5890 . . . . . 6 (𝐹 Fn On → dom 𝐹 = On)
42, 3ax-mp 5 . . . . 5 dom 𝐹 = On
54eleq1i 2679 . . . 4 (dom 𝐹 ∈ V ↔ On ∈ V)
61, 5mtbir 312 . . 3 ¬ dom 𝐹 ∈ V
72tz7.48-2 7402 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
8 funrnex 7004 . . . . . 6 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
98com12 32 . . . . 5 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
10 df-rn 5039 . . . . . 6 ran 𝐹 = dom 𝐹
1110eleq1i 2679 . . . . 5 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
12 dfdm4 5225 . . . . . 6 dom 𝐹 = ran 𝐹
1312eleq1i 2679 . . . . 5 (dom 𝐹 ∈ V ↔ ran 𝐹 ∈ V)
149, 11, 133imtr4g 284 . . . 4 (Fun 𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
157, 14syl 17 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
166, 15mtoi 189 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ ran 𝐹 ∈ V)
172tz7.48-1 7403 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
18 ssexg 4727 . . . 4 ((ran 𝐹𝐴𝐴 ∈ V) → ran 𝐹 ∈ V)
1918ex 449 . . 3 (ran 𝐹𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V))
2017, 19syl 17 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V))
2116, 20mtod 188 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  wss 3540  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  Oncon0 5626  Fun wfun 5784   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798
This theorem is referenced by:  tz7.49  7405
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