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Mirrors > Home > MPE Home > Th. List > csbcnvgALT | Structured version Visualization version GIF version |
Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5228 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
csbcnvgALT | ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbr123 4636 | . . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦) | |
2 | csbconstg 3512 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧) | |
3 | csbconstg 3512 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) | |
4 | 2, 3 | breq12d 4596 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
5 | 1, 4 | syl5bb 271 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)) |
6 | 5 | opabbidv 4648 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}) |
7 | csbopabgALT 4933 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}) | |
8 | df-cnv 5046 | . . . 4 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦} | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}) |
10 | 6, 7, 9 | 3eqtr4rd 2655 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦}) |
11 | df-cnv 5046 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} | |
12 | 11 | csbeq2i 3945 | . 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝑧𝐹𝑦} |
13 | 10, 12 | syl6eqr 2662 | 1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 [wsbc 3402 ⦋csb 3499 class class class wbr 4583 {copab 4642 ◡ccnv 5037 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-cnv 5046 |
This theorem is referenced by: (None) |
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