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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1015 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32179. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1015.1 | ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
bnj1015.2 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1015.13 | ⊢ 𝐷 = (ω ∖ {∅}) |
bnj1015.14 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1015.15 | ⊢ 𝐺 ∈ 𝑉 |
bnj1015.16 | ⊢ 𝐽 ∈ 𝑉 |
Ref | Expression |
---|---|
bnj1015 | ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1015.16 | . . 3 ⊢ 𝐽 ∈ 𝑉 | |
2 | 1 | elexi 3511 | . 2 ⊢ 𝐽 ∈ V |
3 | eleq1 2897 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺 ↔ 𝐽 ∈ dom 𝐺)) | |
4 | 3 | anbi2d 628 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) ↔ (𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺))) |
5 | fveq2 6663 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐺‘𝑗) = (𝐺‘𝐽)) | |
6 | 5 | sseq1d 3995 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
7 | 4, 6 | imbi12d 346 | . 2 ⊢ (𝑗 = 𝐽 → (((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)))) |
8 | bnj1015.15 | . . . 4 ⊢ 𝐺 ∈ 𝑉 | |
9 | 8 | elexi 3511 | . . 3 ⊢ 𝐺 ∈ V |
10 | eleq1 2897 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐵 ↔ 𝐺 ∈ 𝐵)) | |
11 | dmeq 5765 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
12 | 11 | eleq2d 2895 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔 ↔ 𝑗 ∈ dom 𝐺)) |
13 | 10, 12 | anbi12d 630 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺))) |
14 | fveq1 6662 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑗) = (𝐺‘𝑗)) | |
15 | 14 | sseq1d 3995 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
16 | 13, 15 | imbi12d 346 | . . 3 ⊢ (𝑔 = 𝐺 → (((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)))) |
17 | bnj1015.1 | . . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | |
18 | bnj1015.2 | . . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
19 | bnj1015.13 | . . . 4 ⊢ 𝐷 = (ω ∖ {∅}) | |
20 | bnj1015.14 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
21 | 17, 18, 19, 20 | bnj1014 32131 | . . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
22 | 9, 16, 21 | vtocl 3557 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
23 | 2, 7, 22 | vtocl 3557 | 1 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 ∃wrex 3136 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ ciun 4910 dom cdm 5548 suc csuc 6186 Fn wfn 6343 ‘cfv 6348 ωcom 7569 predc-bnj14 31857 trClc-bnj18 31863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-dm 5558 df-iota 6307 df-fv 6356 df-bnj18 31864 |
This theorem is referenced by: bnj1018 32133 |
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