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Mirrors > Home > MPE Home > Th. List > funmoOLD | Structured version Visualization version GIF version |
Description: Obsolete version of funmo 6564 as of 19-Dec-2024. (Contributed by NM, 24-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funmoOLD | ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 6557 | . . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | |
2 | 1 | simplbi 499 | . . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹) |
3 | brrelex1 5730 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V) | |
4 | 3 | ex 414 | . . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V)) |
6 | 5 | ancrd 553 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
7 | 6 | alrimiv 1931 | . 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦))) |
8 | breq1 5152 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦)) | |
9 | 8 | mobidv 2544 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦)) |
10 | 9 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))) |
11 | 1 | simprbi 498 | . . . . . 6 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦) |
12 | 11 | 19.21bi 2183 | . . . . 5 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) |
13 | 10, 12 | vtoclg 3557 | . . . 4 ⊢ (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)) |
14 | 13 | com12 32 | . . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) |
15 | moanimv 2616 | . . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦)) | |
16 | 14, 15 | sylibr 233 | . 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦)) |
17 | moim 2539 | . 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦)) | |
18 | 7, 16, 17 | sylc 65 | 1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∃*wmo 2533 Vcvv 3475 class class class wbr 5149 Rel wrel 5682 Fun wfun 6538 |
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-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-fun 6546 |
This theorem is referenced by: (None) |
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