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Mirrors > Home > MPE Home > Th. List > wfrfun | Structured version Visualization version GIF version |
Description: The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.) |
Ref | Expression |
---|---|
wfrfun.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfrfun | ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wefr 5657 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
3 | weso 5658 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | sopo 5598 | . . . 4 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴) |
7 | simpr 484 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴) | |
8 | wfrfun.1 | . . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
9 | df-wrecs 8293 | . . . 4 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
10 | 8, 9 | eqtri 2752 | . . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
11 | 10 | fprfung 8290 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹) |
12 | 2, 6, 7, 11 | syl3anc 1368 | 1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Po wpo 5577 Or wor 5578 Fr wfr 5619 Se wse 5620 We wwe 5621 ∘ ccom 5671 Fun wfun 6528 2nd c2nd 7968 frecscfrecs 8261 wrecscwrecs 8292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-iota 6486 df-fun 6536 df-fn 6537 df-fv 6542 df-ov 7405 df-frecs 8262 df-wrecs 8293 |
This theorem is referenced by: bpolylem 15994 |
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