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| Description: Obsolete version of wfr2 8376 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 2737 | . . . 4 ⊢ (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) = (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) | |
| 5 | 1, 2, 3, 4 | wfrlem16OLD 8364 | . . 3 ⊢ dom 𝐹 = 𝐴 | 
| 6 | 5 | eleq2i 2833 | . 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴) | 
| 7 | 1, 2, 3 | wfr2aOLD 8366 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) | 
| 8 | 6, 7 | sylbir 235 | 1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 〈cop 4632 Se wse 5635 We wwe 5636 dom cdm 5685 ↾ cres 5687 Predcpred 6320 ‘cfv 6561 wrecscwrecs 8336 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 | 
| This theorem is referenced by: (None) | 
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