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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1520 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 32450. 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 32203 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶) |
4 | 3 | nfcii 2940 | . . . . . 6 ⊢ Ⅎ𝑓𝐶 |
5 | 4 | nfuni 4807 | . . . . 5 ⊢ Ⅎ𝑓∪ 𝐶 |
6 | 1, 5 | nfcxfr 2953 | . . . 4 ⊢ Ⅎ𝑓𝐹 |
7 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑓𝑥 | |
8 | 6, 7 | nffv 6655 | . . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥) |
9 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑓𝐺 | |
10 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅) | |
11 | 6, 10 | nfres 5820 | . . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
12 | 7, 11 | nfop 4781 | . . . 4 ⊢ Ⅎ𝑓〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
13 | 9, 12 | nffv 6655 | . . 3 ⊢ Ⅎ𝑓(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
14 | 8, 13 | nfeq 2968 | . 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 14 | nf5ri 2193 | 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 399 ∀wal 1536 = wceq 1538 {cab 2776 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 〈cop 4531 ∪ cuni 4800 ↾ cres 5521 Fn wfn 6319 ‘cfv 6324 predc-bnj14 32068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-res 5531 df-iota 6283 df-fv 6332 |
This theorem is referenced by: bnj1501 32449 |
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