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Mirrors > Home > MPE Home > Th. List > Mathboxes > setis | Structured version Visualization version GIF version |
Description: Version of setrec2 47226 expressed as an induction schema. This theorem is a generalization of tfis3 7795. (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 4020 | . . . . . . 7 ⊢ (𝑎 ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ 𝑎 𝜓) | |
4 | ssabral 4020 | . . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ (𝐹‘𝑎)𝜓) | |
5 | 3, 4 | imbi12i 351 | . . . . . 6 ⊢ ((𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓}) ↔ (∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
6 | 5 | albii 1822 | . . . . 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 47227 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∣ 𝜓}) |
9 | 8 | sseld 3944 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑏 ∣ 𝜓})) |
10 | setis.2 | . . 3 ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) | |
11 | 10 | elabg 3629 | . 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 1540 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ⊆ wss 3911 ‘cfv 6497 setrecscsetrecs 47214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-setrecs 47215 |
This theorem is referenced by: pgindnf 47247 |
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