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Mirrors > Home > MPE Home > Th. List > Mathboxes > trpredeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
trpredeq3 | ⊢ (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predeq3 5601 | . . . . . 6 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌)) | |
2 | rdgeq2 7395 | . . . . . 6 ⊢ (Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌))) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑋 = 𝑌 → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌))) |
4 | 3 | reseq1d 5316 | . . . 4 ⊢ (𝑋 = 𝑌 → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω)) |
5 | 4 | rneqd 5274 | . . 3 ⊢ (𝑋 = 𝑌 → ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω)) |
6 | 5 | unieqd 4382 | . 2 ⊢ (𝑋 = 𝑌 → ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω)) |
7 | df-trpred 30962 | . 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) | |
8 | df-trpred 30962 | . 2 ⊢ TrPred(𝑅, 𝐴, 𝑌) = ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑌)) ↾ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2669 | 1 ⊢ (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Vcvv 3173 ∪ cuni 4372 ∪ ciun 4455 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 Predcpred 5596 ωcom 6957 reccrdg 7392 TrPredctrpred 30961 |
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-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-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-trpred 30962 |
This theorem is referenced by: trpredeq3d 30969 dftrpred3g 30977 |
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