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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1519 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj1500 30390. 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 |
---|---|
bnj1519.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1519.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1519.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1519.4 | ⊢ 𝐹 = ∪ 𝐶 |
Ref | Expression |
---|---|
bnj1519 | ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1519.4 | . . . . 5 ⊢ 𝐹 = ∪ 𝐶 | |
2 | bnj1519.3 | . . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
3 | nfre1 2988 | . . . . . . . 8 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) | |
4 | 3 | nfab 2755 | . . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
5 | 2, 4 | nfcxfr 2749 | . . . . . 6 ⊢ Ⅎ𝑑𝐶 |
6 | 5 | nfuni 4378 | . . . . 5 ⊢ Ⅎ𝑑∪ 𝐶 |
7 | 1, 6 | nfcxfr 2749 | . . . 4 ⊢ Ⅎ𝑑𝐹 |
8 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑑𝑥 | |
9 | 7, 8 | nffv 6110 | . . 3 ⊢ Ⅎ𝑑(𝐹‘𝑥) |
10 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑑𝐺 | |
11 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅) | |
12 | 7, 11 | nfres 5319 | . . . . 5 ⊢ Ⅎ𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) |
13 | 8, 12 | nfop 4356 | . . . 4 ⊢ Ⅎ𝑑〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
14 | 10, 13 | nffv 6110 | . . 3 ⊢ Ⅎ𝑑(𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
15 | 9, 14 | nfeq 2762 | . 2 ⊢ Ⅎ𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) |
16 | 15 | nf5ri 2053 | 1 ⊢ ((𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉) → ∀𝑑(𝐹‘𝑥) = (𝐺‘〈𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 = wceq 1475 {cab 2596 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 〈cop 4131 ∪ cuni 4372 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 predc-bnj14 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-res 5050 df-iota 5768 df-fv 5812 |
This theorem is referenced by: bnj1501 30389 |
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