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Mirrors > Home > MPE Home > Th. List > Mathboxes > setis | Structured version Visualization version GIF version |
Description: Version of setrec2 47130 expressed as an induction schema. This theorem is a generalization of tfis3 7794. (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 4019 | . . . . . . 7 ⊢ (𝑎 ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ 𝑎 𝜓) | |
4 | ssabral 4019 | . . . . . . 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 1821 | . . . . 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 47131 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∣ 𝜓}) |
9 | 8 | sseld 3943 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑏 ∣ 𝜓})) |
10 | setis.2 | . . 3 ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) | |
11 | 10 | elabg 3628 | . 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 1539 = wceq 1541 ∈ wcel 2106 {cab 2713 ∀wral 3064 ⊆ wss 3910 ‘cfv 6496 setrecscsetrecs 47118 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fv 6504 df-setrecs 47119 |
This theorem is referenced by: pgindnf 47151 |
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