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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj958 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 30332. 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 |
---|---|
bnj958.1 | ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) |
bnj958.2 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
Ref | Expression |
---|---|
bnj958 | ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj958.2 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
2 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
3 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑛 | |
4 | bnj958.1 | . . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
5 | nfiu1 4486 | . . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
6 | 4, 5 | nfcxfr 2749 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐶 |
7 | 3, 6 | nfop 4356 | . . . . . . 7 ⊢ Ⅎ𝑦〈𝑛, 𝐶〉 |
8 | 7 | nfsn 4189 | . . . . . 6 ⊢ Ⅎ𝑦{〈𝑛, 𝐶〉} |
9 | 2, 8 | nfun 3731 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {〈𝑛, 𝐶〉}) |
10 | 1, 9 | nfcxfr 2749 | . . . 4 ⊢ Ⅎ𝑦𝐺 |
11 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑦𝑖 | |
12 | 10, 11 | nffv 6110 | . . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖) |
13 | 12 | nfeq1 2764 | . 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖) |
14 | 13 | nf5ri 2053 | 1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∪ cun 3538 {csn 4125 〈cop 4131 ∪ ciun 4455 ‘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-iun 4457 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: bnj966 30268 bnj967 30269 |
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