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Mirrors > Home > MPE Home > Th. List > wfr2a | Structured version Visualization version GIF version |
Description: A weak version of wfr2 7817 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 6530 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
2 | predeq3 6019 | . . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
3 | 2 | reseq2d 5726 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))) |
4 | 3 | fveq2d 6534 | . . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
5 | 1, 4 | eqeq12d 2808 | . 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 7809 | . 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))) |
10 | 5, 9 | vtoclga 3512 | 1 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 Se wse 5392 We wwe 5393 dom cdm 5435 ↾ cres 5437 Predcpred 6014 ‘cfv 6217 wrecscwrecs 7788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-iota 6181 df-fun 6219 df-fn 6220 df-fv 6225 df-wrecs 7789 |
This theorem is referenced by: wfr2 7817 |
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