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| Mirrors > Home > MPE Home > Th. List > wfr2a | Structured version Visualization version GIF version | ||
| Description: A weak version of wfr2 8376 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 5675 | . . . . 5 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
| 2 | 1 | adantr 480 | . . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
| 3 | weso 5676 | . . . . . 6 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
| 4 | sopo 5611 | . . . . . 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 1129 | . . 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 8337 | . . . . 5 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
| 11 | 9, 10 | eqtri 2765 | . . . 4 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
| 12 | 11 | fpr2a 8327 | . . 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 580 | . 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 8373 | . . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V) |
| 16 | 14, 15 | opco2 8149 | . 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 2777 | 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 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 Po wpo 5590 Or wor 5591 Fr wfr 5634 Se wse 5635 We wwe 5636 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Predcpred 6320 ‘cfv 6561 (class class class)co 7431 2nd c2nd 8013 frecscfrecs 8305 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-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-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-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-fo 6567 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 |
| This theorem is referenced by: wfr2 8376 |
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