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Theorem tz7.48-3 7426
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 6876 . . . 4 ¬ On ∈ V
2 tz7.48.1 . . . . . 6 𝐹 Fn On
3 fndm 5904 . . . . . 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 7424 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
8 funrnex 7026 . . . . . 6 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
98com12 32 . . . . 5 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
10 df-rn 5049 . . . . . 6 ran 𝐹 = dom 𝐹
1110eleq1i 2679 . . . . 5 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
12 dfdm4 5238 . . . . . 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 7425 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
18 ssexg 4732 . . . 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 5037  dom cdm 5038  ran crn 5039  cima 5041  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  cfv 5804
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 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by:  tz7.49  7427
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