| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj958 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. 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 2905 | . . . . . 6 ⊢ Ⅎ𝑦𝑓 | |
| 3 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑦𝑛 | |
| 4 | bnj958.1 | . . . . . . . . 9 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
| 5 | nfiu1 5027 | . . . . . . . . 9 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) | |
| 6 | 4, 5 | nfcxfr 2903 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐶 |
| 7 | 3, 6 | nfop 4889 | . . . . . . 7 ⊢ Ⅎ𝑦〈𝑛, 𝐶〉 |
| 8 | 7 | nfsn 4707 | . . . . . 6 ⊢ Ⅎ𝑦{〈𝑛, 𝐶〉} |
| 9 | 2, 8 | nfun 4170 | . . . . 5 ⊢ Ⅎ𝑦(𝑓 ∪ {〈𝑛, 𝐶〉}) |
| 10 | 1, 9 | nfcxfr 2903 | . . . 4 ⊢ Ⅎ𝑦𝐺 |
| 11 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑦𝑖 | |
| 12 | 10, 11 | nffv 6916 | . . 3 ⊢ Ⅎ𝑦(𝐺‘𝑖) |
| 13 | 12 | nfeq1 2921 | . 2 ⊢ Ⅎ𝑦(𝐺‘𝑖) = (𝑓‘𝑖) |
| 14 | 13 | nf5ri 2195 | 1 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∪ cun 3949 {csn 4626 〈cop 4632 ∪ ciun 4991 ‘cfv 6561 predc-bnj14 34702 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: bnj966 34958 bnj967 34959 |
| Copyright terms: Public domain | W3C validator |