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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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