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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 | onprc 6876 | . . . 4 ⊢ ¬ On ∈ V | |
2 | tz7.48.1 | . . . . . 6 ⊢ 𝐹 Fn On | |
3 | fndm 5904 | . . . . . 6 ⊢ (𝐹 Fn On → dom 𝐹 = On) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ dom 𝐹 = On |
5 | 4 | eleq1i 2679 | . . . 4 ⊢ (dom 𝐹 ∈ V ↔ On ∈ V) |
6 | 1, 5 | mtbir 312 | . . 3 ⊢ ¬ dom 𝐹 ∈ V |
7 | 2 | tz7.48-2 7424 | . . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹) |
8 | funrnex 7026 | . . . . . 6 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V)) | |
9 | 8 | com12 32 | . . . . 5 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V)) |
10 | df-rn 5049 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
11 | 10 | eleq1i 2679 | . . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V) |
12 | dfdm4 5238 | . . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹 | |
13 | 12 | eleq1i 2679 | . . . . 5 ⊢ (dom 𝐹 ∈ V ↔ ran ◡𝐹 ∈ V) |
14 | 9, 11, 13 | 3imtr4g 284 | . . . 4 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
15 | 7, 14 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
16 | 6, 15 | mtoi 189 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V) |
17 | 2 | tz7.48-1 7425 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴) |
18 | ssexg 4732 | . . . 4 ⊢ ((ran 𝐹 ⊆ 𝐴 ∧ 𝐴 ∈ V) → ran 𝐹 ∈ V) | |
19 | 18 | ex 449 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
20 | 17, 19 | syl 17 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
21 | 16, 20 | mtod 188 | 1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 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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