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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem7 | Structured version Visualization version GIF version |
Description: Lemma for founded recursion. The founded recursion generator's domain is a subclass of 𝐴. (Contributed by Scott Fenton, 27-Aug-2022.) |
Ref | Expression |
---|---|
frrlem5.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
frrlem5.2 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
frrlem7 | ⊢ dom 𝐹 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frrlem5.1 | . . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
2 | frrlem5.2 | . . . . . . 7 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | 1, 2 | frrlem5 32677 | . . . . . 6 ⊢ 𝐹 = ∪ 𝐵 |
4 | 3 | dmeqi 5619 | . . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐵 |
5 | dmuni 5629 | . . . . 5 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 | |
6 | 4, 5 | eqtri 2796 | . . . 4 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 |
7 | 6 | sseq1i 3879 | . . 3 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) |
8 | iunss 4831 | . . 3 ⊢ (∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) | |
9 | 7, 8 | bitri 267 | . 2 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) |
10 | 1 | frrlem3 32675 | . 2 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴) |
11 | 9, 10 | mprgbir 3097 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∃wex 1742 {cab 2752 ∀wral 3082 ⊆ wss 3823 ∪ cuni 4708 ∪ ciun 4788 dom cdm 5403 ↾ cres 5405 Predcpred 5982 Fn wfn 6180 ‘cfv 6185 (class class class)co 6974 frecscfrecs 32667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-iota 6149 df-fun 6187 df-fn 6188 df-fv 6193 df-ov 6977 df-frecs 32668 |
This theorem is referenced by: frrlem14 32686 |
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