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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 5388 | . 2 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | eldifvsn 4267 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍)) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (𝑦 ∈ (V ∖ {𝑍}) ↔ 𝑦 ≠ 𝑍) |
5 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
6 | 2, 5 | opelcnv 5226 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) |
7 | df-br 4584 | . . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑅) | |
8 | 6, 7 | bitr4i 266 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ ◡𝑅 ↔ 𝑥𝑅𝑦) |
9 | 4, 8 | anbi12ci 730 | . . . 4 ⊢ ((𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
10 | 9 | exbii 1764 | . . 3 ⊢ (∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)) |
11 | 10 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ (V ∖ {𝑍}) ∧ 〈𝑦, 𝑥〉 ∈ ◡𝑅)} = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
12 | 1, 11 | eqtri 2632 | 1 ⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 {csn 4125 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: suppimacnvss 7192 suppimacnv 7193 |
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