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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1015 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33679. 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 3463 | . 2 ⊢ 𝐽 ∈ V |
3 | eleq1 2822 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ dom 𝐺 ↔ 𝐽 ∈ dom 𝐺)) | |
4 | 3 | anbi2d 630 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) ↔ (𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺))) |
5 | fveq2 6843 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐺‘𝑗) = (𝐺‘𝐽)) | |
6 | 5 | sseq1d 3976 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
7 | 4, 6 | imbi12d 345 | . 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 3463 | . . 3 ⊢ 𝐺 ∈ V |
10 | eleq1 2822 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐵 ↔ 𝐺 ∈ 𝐵)) | |
11 | dmeq 5860 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
12 | 11 | eleq2d 2820 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑗 ∈ dom 𝑔 ↔ 𝑗 ∈ dom 𝐺)) |
13 | 10, 12 | anbi12d 632 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) ↔ (𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺))) |
14 | fveq1 6842 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑗) = (𝐺‘𝑗)) | |
15 | 14 | sseq1d 3976 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
16 | 13, 15 | imbi12d 345 | . . 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 33630 | . . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
22 | 9, 16, 21 | vtocl 3517 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
23 | 2, 7, 22 | vtocl 3517 | 1 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 {csn 4587 ∪ ciun 4955 dom cdm 5634 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 predc-bnj14 33357 trClc-bnj18 33363 |
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 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 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-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-dm 5644 df-iota 6449 df-fv 6505 df-bnj18 33364 |
This theorem is referenced by: bnj1018g 33632 bnj1018 33633 |
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