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Theorem infpssrlem2 10195

Description: Lemma for infpssr 10199. (Contributed by Stefan O'Rear, 30-Oct-2014.)

**Hypotheses**
Ref	Expression
infpssrlem.a	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
infpssrlem.c	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)
infpssrlem.d	⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
infpssrlem.e	⊢ 𝐺 = (rec(^◡𝐹, 𝐶) ↾ ω)

**Assertion**
Ref	Expression
infpssrlem2	⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (^◡𝐹‘(𝐺‘𝑀)))

**Proof of Theorem infpssrlem2**
Step	Hyp	Ref	Expression
1		frsuc 8356	. 2 ⊢ (𝑀 ∈ ω → ((rec(^◡𝐹, 𝐶) ↾ ω)‘suc 𝑀) = (^◡𝐹‘((rec(^◡𝐹, 𝐶) ↾ ω)‘𝑀)))
2		infpssrlem.e	. . 3 ⊢ 𝐺 = (rec(^◡𝐹, 𝐶) ↾ ω)
3	2	fveq1i 6823	. 2 ⊢ (𝐺‘suc 𝑀) = ((rec(^◡𝐹, 𝐶) ↾ ω)‘suc 𝑀)
4	2	fveq1i 6823	. . 3 ⊢ (𝐺‘𝑀) = ((rec(^◡𝐹, 𝐶) ↾ ω)‘𝑀)
5	4	fveq2i 6825	. 2 ⊢ (^◡𝐹‘(𝐺‘𝑀)) = (^◡𝐹‘((rec(^◡𝐹, 𝐶) ↾ ω)‘𝑀))
6	1, 3, 5	3eqtr4g 2791	1 ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (^◡𝐹‘(𝐺‘𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ∖ cdif 3894 ⊆ wss 3897 ^◡ccnv 5613 ↾ cres 5616 suc csuc 6308 –1-1-onto→wf1o 6480 ‘cfv 6481 ωcom 7796 reccrdg 8328

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329

This theorem is referenced by: infpssrlem3 10196 infpssrlem4 10197

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