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Mirrors > Home > MPE Home > Th. List > infpssrlem2 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 10065. (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 8259 | . 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 6772 | . 2 ⊢ (𝐺‘suc 𝑀) = ((rec(◡𝐹, 𝐶) ↾ ω)‘suc 𝑀) |
4 | 2 | fveq1i 6772 | . . 3 ⊢ (𝐺‘𝑀) = ((rec(◡𝐹, 𝐶) ↾ ω)‘𝑀) |
5 | 4 | fveq2i 6774 | . 2 ⊢ (◡𝐹‘(𝐺‘𝑀)) = (◡𝐹‘((rec(◡𝐹, 𝐶) ↾ ω)‘𝑀)) |
6 | 1, 3, 5 | 3eqtr4g 2805 | 1 ⊢ (𝑀 ∈ ω → (𝐺‘suc 𝑀) = (◡𝐹‘(𝐺‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ⊆ wss 3892 ◡ccnv 5589 ↾ cres 5592 suc csuc 6267 –1-1-onto→wf1o 6431 ‘cfv 6432 ωcom 7706 reccrdg 8231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 |
This theorem is referenced by: infpssrlem3 10062 infpssrlem4 10063 |
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