wfrfun - Metamath Proof Explorer

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Theorem wfrfun 8316

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.)

**Hypothesis**
Ref	Expression
wfrfun.1	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

**Assertion**
Ref	Expression
wfrfun	⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

**Proof of Theorem wfrfun**
Step	Hyp	Ref	Expression
1		wefr 5651	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 485	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5652	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5588	. . . 4 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 18	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 485	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 489	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8		wfrfun.1	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9		df-wrecs 8305	. . . 4 ⊢ wrecs(𝑅, 𝐴, 𝐺) = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
10	8, 9	eqtri 2786	. . 3 ⊢ 𝐹 = frecs(𝑅, 𝐴, (𝐺 ∘ 2^nd ))
11	10	fprfung 8302	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
12	2, 6, 7, 11	syl3anc 1398	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 Po wpo 5567 Or wor 5568 Fr wfr 5611 Se wse 5612 We wwe 5613 ∘ ccom 5665 Fun wfun 6530 2^nd c2nd 7981 frecscfrecs 8273 wrecscwrecs 8304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-iota 6492 df-fun 6538 df-fn 6539 df-fv 6544 df-ov 7413 df-frecs 8274 df-wrecs 8305

This theorem is referenced by: bpolylem 16097

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