![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rdgsucmpt2 | Structured version Visualization version GIF version |
Description: This version of rdgsucmpt 8410 avoids the not-free hypothesis of rdgsucmptf 8407 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
rdgsucmpt2.1 | ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
rdgsucmpt2.2 | ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) |
rdgsucmpt2.3 | ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) |
Ref | Expression |
---|---|
rdgsucmpt2 | ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐷 | |
4 | rdgsucmpt2.1 | . . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
5 | rdgsucmpt2.2 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
6 | 5 | cbvmptv 5251 | . . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
7 | rdgeq1 8390 | . . . 4 ⊢ ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
9 | 4, 8 | eqtr4i 2762 | . 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴) |
10 | rdgsucmpt2.3 | . 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) | |
11 | 1, 2, 3, 9, 10 | rdgsucmptf 8407 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3470 ↦ cmpt 5221 Oncon0 6350 suc csuc 6352 ‘cfv 6529 reccrdg 8388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |