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Theorem tz7.48-3 7922
 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 tz7.48.1 . . . . 5 𝐹 Fn On
2 fndm 6317 . . . . 5 (𝐹 Fn On → dom 𝐹 = On)
31, 2ax-mp 5 . . . 4 dom 𝐹 = On
4 onprc 7346 . . . 4 ¬ On ∈ V
53, 4eqneltri 2874 . . 3 ¬ dom 𝐹 ∈ V
61tz7.48-2 7920 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
7 funrnex 7502 . . . . . 6 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
87com12 32 . . . . 5 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
9 df-rn 5446 . . . . . 6 ran 𝐹 = dom 𝐹
109eleq1i 2871 . . . . 5 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
11 dfdm4 5642 . . . . . 6 dom 𝐹 = ran 𝐹
1211eleq1i 2871 . . . . 5 (dom 𝐹 ∈ V ↔ ran 𝐹 ∈ V)
138, 10, 123imtr4g 297 . . . 4 (Fun 𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
146, 13syl 17 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
155, 14mtoi 200 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ ran 𝐹 ∈ V)
161tz7.48-1 7921 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
17 ssexg 5111 . . . 4 ((ran 𝐹𝐴𝐴 ∈ V) → ran 𝐹 ∈ V)
1817ex 413 . . 3 (ran 𝐹𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V))
1916, 18syl 17 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V))
2015, 19mtod 199 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1520   ∈ wcel 2079  ∀wral 3103  Vcvv 3432   ∖ cdif 3851   ⊆ wss 3854  ◡ccnv 5434  dom cdm 5435  ran crn 5436   “ cima 5438  Oncon0 6058  Fun wfun 6211   Fn wfn 6212  ‘cfv 6217 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pr 5214  ax-un 7310 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-ord 6061  df-on 6062  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225 This theorem is referenced by:  tz7.49  7923
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