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Mirrors > Home > MPE Home > Th. List > wfr2a | Structured version Visualization version GIF version |
Description: A weak version of wfr2 8074 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.) |
Ref | Expression |
---|---|
wfr2a.1 | ⊢ 𝑅 We 𝐴 |
wfr2a.2 | ⊢ 𝑅 Se 𝐴 |
wfr2a.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr2a | ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6717 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
2 | predeq3 6164 | . . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
3 | 2 | reseq2d 5851 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))) |
4 | 3 | fveq2d 6721 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
5 | 1, 4 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) ↔ (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))) |
6 | wfr2a.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
7 | wfr2a.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
8 | wfr2a.3 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
9 | 6, 7, 8 | wfrlem12 8066 | . 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))) |
10 | 5, 9 | vtoclga 3489 | 1 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Se wse 5507 We wwe 5508 dom cdm 5551 ↾ cres 5553 Predcpred 6159 ‘cfv 6380 wrecscwrecs 8046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-iota 6338 df-fun 6382 df-fn 6383 df-fv 6388 df-wrecs 8047 |
This theorem is referenced by: wfr2 8074 |
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