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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 | 1 | fndmi 6604 | . . . 4 ⊢ dom 𝐹 = On |
3 | onprc 7705 | . . . 4 ⊢ ¬ On ∈ V | |
4 | 2, 3 | eqneltri 2832 | . . 3 ⊢ ¬ dom 𝐹 ∈ V |
5 | 1 | tz7.48-2 8381 | . . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹) |
6 | funrnex 7879 | . . . . . 6 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V)) | |
7 | 6 | com12 32 | . . . . 5 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V)) |
8 | df-rn 5643 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
9 | 8 | eleq1i 2829 | . . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V) |
10 | dfdm4 5850 | . . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹 | |
11 | 10 | eleq1i 2829 | . . . . 5 ⊢ (dom 𝐹 ∈ V ↔ ran ◡𝐹 ∈ V) |
12 | 7, 9, 11 | 3imtr4g 296 | . . . 4 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
13 | 5, 12 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V)) |
14 | 4, 13 | mtoi 198 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V) |
15 | 1 | tz7.48-1 8382 | . . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴) |
16 | ssexg 5279 | . . . 4 ⊢ ((ran 𝐹 ⊆ 𝐴 ∧ 𝐴 ∈ V) → ran 𝐹 ∈ V) | |
17 | 16 | ex 414 | . . 3 ⊢ (ran 𝐹 ⊆ 𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
18 | 15, 17 | syl 17 | . 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V)) |
19 | 14, 18 | mtod 197 | 1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 ∀wral 3063 Vcvv 3444 ∖ cdif 3906 ⊆ wss 3909 ◡ccnv 5631 dom cdm 5632 ran crn 5633 “ cima 5635 Oncon0 6316 Fun wfun 6488 Fn wfn 6489 ‘cfv 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 |
This theorem is referenced by: tz7.49 8384 |
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