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Mirrors > Home > MPE Home > Th. List > predeq123 | Structured version Visualization version GIF version |
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
predeq123 | ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1055 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → 𝐴 = 𝐵) | |
2 | cnveq 5218 | . . . . 5 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆) | |
3 | 2 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → ◡𝑅 = ◡𝑆) |
4 | sneq 4135 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
5 | 4 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → {𝑋} = {𝑌}) |
6 | 3, 5 | imaeq12d 5386 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (◡𝑅 “ {𝑋}) = (◡𝑆 “ {𝑌})) |
7 | 1, 6 | ineq12d 3777 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → (𝐴 ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑆 “ {𝑌}))) |
8 | df-pred 5597 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
9 | df-pred 5597 | . 2 ⊢ Pred(𝑆, 𝐵, 𝑌) = (𝐵 ∩ (◡𝑆 “ {𝑌})) | |
10 | 7, 8, 9 | 3eqtr4g 2669 | 1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∩ cin 3539 {csn 4125 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
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-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-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 df-pred 5597 |
This theorem is referenced by: predeq1 5599 predeq2 5600 predeq3 5601 wsuceq123 31004 wlimeq12 31009 |
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