wfrfun - Metamath Proof Explorer

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Theorem wfrfun 8366

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 5645	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
2	1	adantr 486	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3		weso 5646	. . . 4 ⊢ (𝑅 We 𝐴 → 𝑅 Or 𝐴)
4		sopo 5582	. . . 4 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
5	3, 4	syl 18	. . 3 ⊢ (𝑅 We 𝐴 → 𝑅 Po 𝐴)
6	5	adantr 486	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7		simpr 490	. 2 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8		wfrfun.1	. . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9		df-wrecs 8355	. . . 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 8352	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
12	2, 6, 7, 11	syl3anc 1399	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 Po wpo 5561 Or wor 5562 Fr wfr 5605 Se wse 5606 We wwe 5607 ∘ ccom 5659 Fun wfun 6549 2^nd c2nd 8021 frecscfrecs 8323 wrecscwrecs 8354

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-iota 6509 df-fun 6557 df-fn 6558 df-fv 6563 df-ov 7443 df-frecs 8324 df-wrecs 8355

This theorem is used by: bpolylem 16240

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