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Mirrors > Home > MPE Home > Th. List > wfr2OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of wfr2 8282 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfr2OLD.1 | ⊢ 𝑅 We 𝐴 |
wfr2OLD.2 | ⊢ 𝑅 Se 𝐴 |
wfr2OLD.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr2OLD | ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfr2OLD.1 | . . . 4 ⊢ 𝑅 We 𝐴 | |
2 | wfr2OLD.2 | . . . 4 ⊢ 𝑅 Se 𝐴 | |
3 | wfr2OLD.3 | . . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | eqid 2736 | . . . 4 ⊢ (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) = (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) | |
5 | 1, 2, 3, 4 | wfrlem16OLD 8270 | . . 3 ⊢ dom 𝐹 = 𝐴 |
6 | 5 | eleq2i 2829 | . 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴) |
7 | 1, 2, 3 | wfr2aOLD 8272 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
8 | 6, 7 | sylbir 234 | 1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∪ cun 3908 {csn 4586 〈cop 4592 Se wse 5586 We wwe 5587 dom cdm 5633 ↾ cres 5635 Predcpred 6252 ‘cfv 6496 wrecscwrecs 8242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-2nd 7922 df-frecs 8212 df-wrecs 8243 |
This theorem is referenced by: (None) |
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