![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1123 | 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 |
---|---|
bnj1123.4 | ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
bnj1123.3 | ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj1123.1 | ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) |
bnj1123.2 | ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂) |
Ref | Expression |
---|---|
bnj1123 | ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1123.2 | . 2 ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂) | |
2 | bnj1123.1 | . . 3 ⊢ (𝜂 ↔ ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) | |
3 | 2 | sbcbii 3800 | . 2 ⊢ ([𝑗 / 𝑖]𝜂 ↔ [𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵)) |
4 | bnj1123.3 | . . . . . . . 8 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
5 | nfcv 2904 | . . . . . . . . . 10 ⊢ Ⅎ𝑖𝐷 | |
6 | nfv 1918 | . . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑓 Fn 𝑛 | |
7 | nfv 1918 | . . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑 | |
8 | bnj1123.4 | . . . . . . . . . . . . 13 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) | |
9 | 8 | bnj1095 33450 | . . . . . . . . . . . 12 ⊢ (𝜓 → ∀𝑖𝜓) |
10 | 9 | nf5i 2143 | . . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜓 |
11 | 6, 7, 10 | nf3an 1905 | . . . . . . . . . 10 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) |
12 | 5, 11 | nfrexw 3295 | . . . . . . . . 9 ⊢ Ⅎ𝑖∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) |
13 | 12 | nfab 2910 | . . . . . . . 8 ⊢ Ⅎ𝑖{𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
14 | 4, 13 | nfcxfr 2902 | . . . . . . 7 ⊢ Ⅎ𝑖𝐾 |
15 | 14 | nfcri 2891 | . . . . . 6 ⊢ Ⅎ𝑖 𝑓 ∈ 𝐾 |
16 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑖 𝑗 ∈ dom 𝑓 | |
17 | 15, 16 | nfan 1903 | . . . . 5 ⊢ Ⅎ𝑖(𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) |
18 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑖(𝑓‘𝑗) ⊆ 𝐵 | |
19 | 17, 18 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑖((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵) |
20 | eleq1w 2817 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓 ↔ 𝑗 ∈ dom 𝑓)) | |
21 | 20 | anbi2d 630 | . . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓))) |
22 | fveq2 6843 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗)) | |
23 | 22 | sseq1d 3976 | . . . . 5 ⊢ (𝑖 = 𝑗 → ((𝑓‘𝑖) ⊆ 𝐵 ↔ (𝑓‘𝑗) ⊆ 𝐵)) |
24 | 21, 23 | imbi12d 345 | . . . 4 ⊢ (𝑖 = 𝑗 → (((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))) |
25 | 19, 24 | sbciegf 3779 | . . 3 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))) |
26 | 25 | elv 3450 | . 2 ⊢ ([𝑗 / 𝑖]((𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓) → (𝑓‘𝑖) ⊆ 𝐵) ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)) |
27 | 1, 3, 26 | 3bitri 297 | 1 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵)) |
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 Vcvv 3444 [wsbc 3740 ⊆ wss 3911 ∪ ciun 4955 dom cdm 5634 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 predc-bnj14 33357 |
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-sbc 3741 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-br 5107 df-iota 6449 df-fv 6505 |
This theorem is referenced by: bnj1030 33656 |
Copyright terms: Public domain | W3C validator |