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Theorem tz7.48-3 7745
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 7184 . . . 4 ¬ On ∈ V
2 tz7.48.1 . . . . . 6 𝐹 Fn On
3 fndm 6170 . . . . . 6 (𝐹 Fn On → dom 𝐹 = On)
42, 3ax-mp 5 . . . . 5 dom 𝐹 = On
54eleq1i 2835 . . . 4 (dom 𝐹 ∈ V ↔ On ∈ V)
61, 5mtbir 314 . . 3 ¬ dom 𝐹 ∈ V
72tz7.48-2 7743 . . . 4 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → Fun 𝐹)
8 funrnex 7333 . . . . . 6 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
98com12 32 . . . . 5 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
10 df-rn 5290 . . . . . 6 ran 𝐹 = dom 𝐹
1110eleq1i 2835 . . . . 5 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
12 dfdm4 5486 . . . . . 6 dom 𝐹 = ran 𝐹
1312eleq1i 2835 . . . . 5 (dom 𝐹 ∈ V ↔ ran 𝐹 ∈ V)
149, 11, 133imtr4g 287 . . . 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 190 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ ran 𝐹 ∈ V)
172tz7.48-1 7744 . . 3 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ran 𝐹𝐴)
18 ssexg 4967 . . . 4 ((ran 𝐹𝐴𝐴 ∈ V) → ran 𝐹 ∈ V)
1918ex 401 . . 3 (ran 𝐹𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V))
2017, 19syl 17 . 2 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V))
2116, 20mtod 189 1 (∀𝑥 ∈ On (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) → ¬ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  wss 3734  ccnv 5278  dom cdm 5279  ran crn 5280  cima 5282  Oncon0 5910  Fun wfun 6064   Fn wfn 6065  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078
This theorem is referenced by:  tz7.49  7746
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