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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 5899 | . 2 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} | |
2 | eldifvsn 4690 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍)) | |
3 | 2 | elv 3446 | . . . . 5 ⊢ (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍) |
4 | vex 3444 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
5 | vex 3444 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 4, 5 | opelcnv 5716 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | df-br 5031 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
8 | 6, 7 | bitr4i 281 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 𝑥𝑅𝑦) |
9 | 3, 8 | anbi12ci 630 | . . . 4 ⊢ ((𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
10 | 9 | exbii 1849 | . . 3 ⊢ (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
11 | 10 | abbii 2863 | . 2 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
12 | 1, 11 | eqtri 2821 | 1 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 {csn 4525 〈cop 4531 class class class wbr 5030 ◡ccnv 5518 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: suppimacnvss 7823 suppimacnv 7824 |
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