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Mirrors > Home > MPE Home > Th. List > wfr2a | Structured version Visualization version GIF version |
Description: A weak version of wfr2 8357 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.) (Proof shortened by Scott Fenton, 18-Nov-2024.) |
Ref | Expression |
---|---|
wfrfun.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr2a | ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 5668 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
3 | weso 5669 | . . . . . 6 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | sopo 5609 | . . . . . 6 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴) |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴) |
7 | simpr 484 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴) | |
8 | 2, 6, 7 | 3jca 1126 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴)) |
9 | wfrfun.1 | . . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
10 | df-wrecs 8318 | . . . . 5 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
11 | 9, 10 | eqtri 2756 | . . . 4 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
12 | 11 | fpr2a 8308 | . . 3 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋(𝐺 ∘ 2nd )(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
13 | 8, 12 | sylan 579 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋(𝐺 ∘ 2nd )(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
14 | simpr 484 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → 𝑋 ∈ dom 𝐹) | |
15 | 9 | wfrresex 8354 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V) |
16 | 14, 15 | opco2 8129 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝑋(𝐺 ∘ 2nd )(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
17 | 13, 16 | eqtrd 2768 | 1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 Po wpo 5588 Or wor 5589 Fr wfr 5630 Se wse 5631 We wwe 5632 dom cdm 5678 ↾ cres 5680 ∘ ccom 5682 Predcpred 6304 ‘cfv 6548 (class class class)co 7420 2nd c2nd 7992 frecscfrecs 8286 wrecscwrecs 8317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-ov 7423 df-2nd 7994 df-frecs 8287 df-wrecs 8318 |
This theorem is referenced by: wfr2 8357 |
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