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Mirrors > Home > MPE Home > Th. List > Mathboxes > setis | Structured version Visualization version GIF version |
Description: Version of setrec2 44791 expressed as an induction schema. This theorem is a generalization of tfis3 7566. (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 4042 | . . . . . . 7 ⊢ (𝑎 ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ 𝑎 𝜓) | |
4 | ssabral 4042 | . . . . . . 7 ⊢ ((𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓} ↔ ∀𝑏 ∈ (𝐹‘𝑎)𝜓) | |
5 | 3, 4 | imbi12i 353 | . . . . . 6 ⊢ ((𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓}) ↔ (∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
6 | 5 | albii 1816 | . . . . 5 ⊢ (∀𝑎(𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓}) ↔ ∀𝑎(∀𝑏 ∈ 𝑎 𝜓 → ∀𝑏 ∈ (𝐹‘𝑎)𝜓)) |
7 | 2, 6 | sylibr 236 | . . . 4 ⊢ (𝜑 → ∀𝑎(𝑎 ⊆ {𝑏 ∣ 𝜓} → (𝐹‘𝑎) ⊆ {𝑏 ∣ 𝜓})) |
8 | 1, 7 | setrec2v 44792 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∣ 𝜓}) |
9 | 8 | sseld 3966 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝑏 ∣ 𝜓})) |
10 | setis.2 | . . 3 ⊢ (𝑏 = 𝐴 → (𝜓 ↔ 𝜒)) | |
11 | 10 | elabg 3666 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑏 ∣ 𝜓} ↔ 𝜒)) |
12 | 9, 11 | mpbidi 243 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ⊆ wss 3936 ‘cfv 6350 setrecscsetrecs 44779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fv 6358 df-setrecs 44780 |
This theorem is referenced by: (None) |
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