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Mirrors > Home > MPE Home > Th. List > Mathboxes > setis | Structured version Visualization version GIF version |
Description: Version of setrec2 47987 expressed as an induction schema. This theorem is a generalization of tfis3 7841. (Contributed by Emmett Weisz, 27-Feb-2022.) |
Ref | Expression |
---|---|
setis.1 | ⊢ 𝐵 = setrecs(𝐹) |
setis.2 | ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) |
setis.3 | ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
Ref | Expression |
---|---|
setis | ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setis.1 | . . . 4 ⊢ 𝐵 = setrecs(𝐹) | |
2 | setis.3 | . . . . 5 ⊢ (𝜑 → ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) | |
3 | ssabral 4052 | . . . . . . 7 ⊢ (𝑎 ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ 𝑎 𝜓) | |
4 | ssabral 4052 | . . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ (𝐹‘𝑎)𝜓) | |
5 | 3, 4 | imbi12i 350 | . . . . . 6 ⊢ ((𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓}) ↔ (∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
6 | 5 | albii 1813 | . . . . 5 ⊢ (∀𝑎(𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓}) ↔ ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
7 | 2, 6 | sylibr 233 | . . . 4 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓})) |
8 | 1, 7 | setrec2v 47988 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∣ 𝜓}) |
9 | 8 | sseld 3974 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑏 ∣ 𝜓})) |
10 | setis.2 | . . 3 ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) | |
11 | 10 | elabg 3659 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑏 ∣ 𝜓} ↔ 𝜒)) |
12 | 9, 11 | mpbidi 240 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2701 ∀wral 3053 ⊆ wss 3941 ‘cfv 6534 setrecscsetrecs 47975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-setrecs 47976 |
This theorem is referenced by: pgindnf 48008 |
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