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Mirrors > Home > MPE Home > Th. List > infpssrlem1 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 9947. (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 |
---|---|
infpssrlem1 | ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infpssrlem.e | . . 3 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) | |
2 | 1 | fveq1i 6737 | . 2 ⊢ (𝐺‘∅) = ((rec(◡𝐹, 𝐶) ↾ ω)‘∅) |
3 | infpssrlem.d | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵)) | |
4 | fr0g 8192 | . . 3 ⊢ (𝐶 ∈ (𝐴 ∖ 𝐵) → ((rec(◡𝐹, 𝐶) ↾ ω)‘∅) = 𝐶) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ((rec(◡𝐹, 𝐶) ↾ ω)‘∅) = 𝐶) |
6 | 2, 5 | eqtrid 2790 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4252 ◡ccnv 5565 ↾ cres 5568 –1-1-onto→wf1o 6397 ‘cfv 6398 ωcom 7663 reccrdg 8166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-om 7664 df-wrecs 8068 df-recs 8129 df-rdg 8167 |
This theorem is referenced by: infpssrlem3 9944 infpssrlem4 9945 |
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