![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elfvov1 | Structured version Visualization version GIF version |
Description: Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 18-May-2025.) |
Ref | Expression |
---|---|
elfvov1.o | ⊢ Rel dom 𝑂 |
elfvov1.s | ⊢ 𝑆 = (𝐼𝑂𝑅) |
elfvov1.x | ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌)) |
Ref | Expression |
---|---|
elfvov1 | ⊢ (𝜑 → 𝐼 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvov1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌)) | |
2 | n0i 4346 | . . 3 ⊢ (𝑋 ∈ (𝑆‘𝑌) → ¬ (𝑆‘𝑌) = ∅) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ¬ (𝑆‘𝑌) = ∅) |
4 | elfvov1.s | . . . . 5 ⊢ 𝑆 = (𝐼𝑂𝑅) | |
5 | elfvov1.o | . . . . . 6 ⊢ Rel dom 𝑂 | |
6 | 5 | ovprc1 7470 | . . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼𝑂𝑅) = ∅) |
7 | 4, 6 | eqtrid 2787 | . . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑆 = ∅) |
8 | 7 | fveq1d 6909 | . . 3 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = (∅‘𝑌)) |
9 | 0fv 6951 | . . 3 ⊢ (∅‘𝑌) = ∅ | |
10 | 8, 9 | eqtrdi 2791 | . 2 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = ∅) |
11 | 3, 10 | nsyl2 141 | 1 ⊢ (𝜑 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 dom cdm 5689 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: ismhp 22162 mhprcl 22165 mhpmulcl 22171 mhppwdeg 22172 mhpaddcl 22173 mhpinvcl 22174 mhpvscacl 22176 mhpind 42581 evlsmhpvvval 42582 mhphf2 42585 mhphf3 42586 |
Copyright terms: Public domain | W3C validator |