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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1015 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32286. 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 3516 | . 2 ⊢ 𝐽 ∈ V |
3 | eleq1 2903 | . . . 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 6673 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐺‘𝑗) = (𝐺‘𝐽)) | |
6 | 5 | sseq1d 4001 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
7 | 4, 6 | imbi12d 347 | . 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 3516 | . . 3 ⊢ 𝐺 ∈ V |
10 | eleq1 2903 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ 𝐵 ↔ 𝐺 ∈ 𝐵)) | |
11 | dmeq 5775 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
12 | 11 | eleq2d 2901 | . . . . 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 6672 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑗) = (𝐺‘𝑗)) | |
15 | 14 | sseq1d 4001 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅))) |
16 | 13, 15 | imbi12d 347 | . . 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 32237 | . . 3 ⊢ ((𝑔 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝑔) → (𝑔‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
22 | 9, 16, 21 | vtocl 3562 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝑗 ∈ dom 𝐺) → (𝐺‘𝑗) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
23 | 2, 7, 22 | vtocl 3562 | 1 ⊢ ((𝐺 ∈ 𝐵 ∧ 𝐽 ∈ dom 𝐺) → (𝐺‘𝐽) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {cab 2802 ∀wral 3141 ∃wrex 3142 ∖ cdif 3936 ⊆ wss 3939 ∅c0 4294 {csn 4570 ∪ ciun 4922 dom cdm 5558 suc csuc 6196 Fn wfn 6353 ‘cfv 6358 ωcom 7583 predc-bnj14 31962 trClc-bnj18 31968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-bnj18 31969 |
This theorem is referenced by: bnj1018g 32239 bnj1018 32240 |
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