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Mirrors > Home > MPE Home > Th. List > tz7.48-3 | Structured version Visualization version GIF version |
Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) |
Ref | Expression |
---|---|
tz7.48.1 | ⊢ 𝐹 Fn On |
Ref | Expression |
---|---|
tz7.48-3 | ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tz7.48.1 | . . . . 5 ⊢ 𝐹 Fn On | |
2 | fndm 6455 | . . . . 5 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ dom 𝐹 = On |
4 | onprc 7499 | . . . 4 ⊢ ¬ On ∈ V | |
5 | 3, 4 | eqneltri 2906 | . . 3 ⊢ ¬ dom 𝐹 ∈ V |
6 | 1 | tz7.48-2 8078 | . . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹) |
7 | funrnex 7655 | . . . . . 6 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V)) | |
8 | 7 | com12 32 | . . . . 5 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V)) |
9 | df-rn 5566 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
10 | 9 | eleq1i 2903 | . . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V) |
11 | dfdm4 5764 | . . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹 | |
12 | 11 | eleq1i 2903 | . . . . 5 ⊢ (dom 𝐹 ∈ V ↔ ran ◡𝐹 ∈ V) |
13 | 8, 10, 12 | 3imtr4g 298 | . . . 4 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
14 | 6, 13 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
15 | 5, 14 | mtoi 201 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V) |
16 | 1 | tz7.48-1 8079 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴) |
17 | ssexg 5227 | . . . 4 ⊢ ((ran 𝐹 ⊆ 𝐴 ∧ 𝐴 ∈ V) → ran 𝐹 ∈ V) | |
18 | 17 | ex 415 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
19 | 16, 18 | syl 17 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
20 | 15, 19 | mtod 200 | 1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ∖ cdif 3933 ⊆ wss 3936 ◡ccnv 5554 dom cdm 5555 ran crn 5556 “ cima 5558 Oncon0 6191 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 |
This theorem is referenced by: tz7.49 8081 |
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