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Theorem bnj229 30697
Description: Technical lemma for bnj517 30698. 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.)
Hypothesis
Ref Expression
bnj229.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj229 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑖,𝐹,𝑚,𝑦   𝑖,𝑁,𝑚   𝑅,𝑖,𝑚
Allowed substitution hints:   𝜓(𝑦,𝑖,𝑚,𝑛)   𝐴(𝑛)   𝑅(𝑦,𝑛)   𝐹(𝑛)   𝑁(𝑦,𝑛)

Proof of Theorem bnj229
StepHypRef Expression
1 bnj213 30695 . . 3 pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
21bnj226 30545 . 2 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
3 bnj229.1 . . . . . . . 8 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑁 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
43bnj222 30696 . . . . . . 7 (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
54bnj228 30546 . . . . . 6 ((𝑚 ∈ ω ∧ 𝜓) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
65adantl 482 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
7 eleq1 2686 . . . . . . 7 (suc 𝑚 = 𝑛 → (suc 𝑚𝑁𝑛𝑁))
8 fveq2 6153 . . . . . . . 8 (suc 𝑚 = 𝑛 → (𝐹‘suc 𝑚) = (𝐹𝑛))
98eqeq1d 2623 . . . . . . 7 (suc 𝑚 = 𝑛 → ((𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
107, 9imbi12d 334 . . . . . 6 (suc 𝑚 = 𝑛 → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
1110adantr 481 . . . . 5 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → ((suc 𝑚𝑁 → (𝐹‘suc 𝑚) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)) ↔ (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))))
126, 11mpbid 222 . . . 4 ((suc 𝑚 = 𝑛 ∧ (𝑚 ∈ ω ∧ 𝜓)) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
13123impb 1257 . . 3 ((suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓) → (𝑛𝑁 → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅)))
1413impcom 446 . 2 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) = 𝑦 ∈ (𝐹𝑚) pred(𝑦, 𝐴, 𝑅))
152, 14bnj1262 30624 1 ((𝑛𝑁 ∧ (suc 𝑚 = 𝑛𝑚 ∈ ω ∧ 𝜓)) → (𝐹𝑛) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3559   ciun 4490  suc csuc 5689  cfv 5852  ωcom 7019   predc-bnj14 30496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-suc 5693  df-iota 5815  df-fv 5860  df-bnj14 30497
This theorem is referenced by: (None)
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