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Mirrors > Home > MPE Home > Th. List > cnvimadfsn | Structured version Visualization version GIF version |
Description: The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.) |
Ref | Expression |
---|---|
cnvimadfsn | ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfima3 5966 | . 2 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} | |
2 | eldifvsn 4731 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍)) | |
3 | 2 | elv 3436 | . . . . 5 ⊢ (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍) |
4 | vex 3434 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | vex 3434 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | opelcnv 5784 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | df-br 5075 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
8 | 6, 7 | bitr4i 277 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 𝑥𝑅𝑦) |
9 | 3, 8 | anbi12ci 628 | . . . 4 ⊢ ((𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
10 | 9 | exbii 1850 | . . 3 ⊢ (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
11 | 10 | abbii 2808 | . 2 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
12 | 1, 11 | eqtri 2766 | 1 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ≠ wne 2943 Vcvv 3430 ∖ cdif 3884 {csn 4562 〈cop 4568 class class class wbr 5074 ◡ccnv 5584 “ cima 5588 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-br 5075 df-opab 5137 df-xp 5591 df-cnv 5593 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 |
This theorem is referenced by: suppimacnvss 7977 suppimacnv 7978 |
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