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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 5668 | . . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴) |
3 | weso 5669 | . . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴) | |
4 | sopo 5609 | . . . 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 8318 | . . . 4 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) | |
10 | 8, 9 | eqtri 2756 | . . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2nd )) |
11 | 10 | fprfung 8315 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹) |
12 | 2, 6, 7, 11 | syl3anc 1369 | 1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Po wpo 5588 Or wor 5589 Fr wfr 5630 Se wse 5631 We wwe 5632 ∘ ccom 5682 Fun wfun 6542 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 |
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-fv 6556 df-ov 7423 df-frecs 8287 df-wrecs 8318 |
This theorem is referenced by: bpolylem 16025 |
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