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Mirrors > Home > MPE Home > Th. List > infpssrlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 9530. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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) |
Ref | Expression |
---|---|
infpssrlem2 | ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (◡𝐹‘(𝐺‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frsuc 7878 | . 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 6502 | . 2 ⊢ (𝐺‘suc 𝑀) = ((rec(◡𝐹, 𝐶) ↾ ω)‘suc 𝑀) |
4 | 2 | fveq1i 6502 | . . 3 ⊢ (𝐺‘𝑀) = ((rec(◡𝐹, 𝐶) ↾ ω)‘𝑀) |
5 | 4 | fveq2i 6504 | . 2 ⊢ (◡𝐹‘(𝐺‘𝑀)) = (◡𝐹‘((rec(◡𝐹, 𝐶) ↾ ω)‘𝑀)) |
6 | 1, 3, 5 | 3eqtr4g 2839 | 1 ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (◡𝐹‘(𝐺‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∖ cdif 3828 ⊆ wss 3831 ◡ccnv 5407 ↾ cres 5410 suc csuc 6033 –1-1-onto→wf1o 6189 ‘cfv 6190 ωcom 7398 reccrdg 7851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 |
This theorem is referenced by: infpssrlem3 9527 infpssrlem4 9528 |
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