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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1520 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 32342. 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 |
---|---|
bnj1520.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1520.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1520.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1520.4 | ⊢ 𝐹 = ∪ 𝐶 |
Ref | Expression |
---|---|
bnj1520 | ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1520.4 | . . . . 5 ⊢ 𝐹 = ∪ 𝐶 | |
2 | bnj1520.3 | . . . . . . . 8 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
3 | 2 | bnj1317 32095 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
4 | 3 | nfcii 2967 | . . . . . 6 ⊢ Ⅎ𝑓𝐶 |
5 | 4 | nfuni 4847 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐶 |
6 | 1, 5 | nfcxfr 2977 | . . . 4 ⊢ Ⅎ𝑓𝐹 |
7 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑓𝑥 | |
8 | 6, 7 | nffv 6682 | . . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥) |
9 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑓𝐺 | |
10 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
11 | 6, 10 | nfres 5857 | . . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
12 | 7, 11 | nfop 4821 | . . . 4 ⊢ Ⅎ𝑓〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
13 | 9, 12 | nffv 6682 | . . 3 ⊢ Ⅎ𝑓(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
14 | 8, 13 | nfeq 2993 | . 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 14 | nf5ri 2195 | 1 ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 = wceq 1537 {cab 2801 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 〈cop 4575 ∪ cuni 4840 ↾ cres 5559 Fn wfn 6352 ‘cfv 6357 predc-bnj14 31960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-res 5569 df-iota 6316 df-fv 6365 |
This theorem is referenced by: bnj1501 32341 |
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