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Mirrors > Home > MPE Home > Th. List > wfr2aOLD | Structured version Visualization version GIF version |
Description: Obsolete version of wfr2a 8335 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 30-Jul-2020.) |
Ref | Expression |
---|---|
wfr2aOLD.1 | ⊢ 𝑅 We 𝐴 |
wfr2aOLD.2 | ⊢ 𝑅 Se 𝐴 |
wfr2aOLD.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr2aOLD | ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6885 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
2 | predeq3 6298 | . . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
3 | 2 | reseq2d 5975 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))) |
4 | 3 | fveq2d 6889 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
5 | 1, 4 | eqeq12d 2742 | . 2 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) ↔ (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))) |
6 | wfr2aOLD.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
7 | wfr2aOLD.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
8 | wfr2aOLD.3 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
9 | 6, 7, 8 | wfrlem12OLD 8321 | . 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))) |
10 | 5, 9 | vtoclga 3560 | 1 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Se wse 5622 We wwe 5623 dom cdm 5669 ↾ cres 5671 Predcpred 6293 ‘cfv 6537 wrecscwrecs 8297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-ov 7408 df-2nd 7975 df-frecs 8267 df-wrecs 8298 |
This theorem is referenced by: wfr2OLD 8329 |
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