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Mirrors > Home > ILE Home > Th. List > cnveq | GIF version |
Description: Equality theorem for converse. (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
cnveq | ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 4818 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | cnvss 4818 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ◡𝐵 ⊆ ◡𝐴) | |
3 | 1, 2 | anim12i 338 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (◡𝐴 ⊆ ◡𝐵 ∧ ◡𝐵 ⊆ ◡𝐴)) |
4 | eqss 3185 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3185 | . 2 ⊢ (◡𝐴 = ◡𝐵 ↔ (◡𝐴 ⊆ ◡𝐵 ∧ ◡𝐵 ⊆ ◡𝐴)) | |
6 | 3, 4, 5 | 3imtr4i 201 | 1 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ⊆ wss 3144 ◡ccnv 4643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-in 3150 df-ss 3157 df-br 4019 df-opab 4080 df-cnv 4652 |
This theorem is referenced by: cnveqi 4820 cnveqd 4821 rneq 4872 cnveqb 5102 funcnvuni 5304 f1eq1 5435 f1o00 5515 foeqcnvco 5811 tposfn2 6290 ereq1 6565 infeq3 7043 1arith 12398 isrim0 13508 psrbag 13944 iscn 14149 ishmeo 14256 |
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