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Theorem isf32lem1 10348
Description: Lemma for isfin3-2 10362. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a (πœ‘ β†’ 𝐹:Ο‰βŸΆπ’« 𝐺)
isf32lem.b (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
isf32lem.c (πœ‘ β†’ Β¬ ∩ ran 𝐹 ∈ ran 𝐹)
Assertion
Ref Expression
isf32lem1 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 βŠ† 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐴   π‘₯,𝐹
Allowed substitution hint:   𝐺(π‘₯)

Proof of Theorem isf32lem1
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . 5 (π‘Ž = 𝐡 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΅))
21sseq1d 4014 . . . 4 (π‘Ž = 𝐡 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)))
32imbi2d 341 . . 3 (π‘Ž = 𝐡 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅))))
4 fveq2 6892 . . . . 5 (π‘Ž = 𝑏 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π‘))
54sseq1d 4014 . . . 4 (π‘Ž = 𝑏 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅)))
65imbi2d 341 . . 3 (π‘Ž = 𝑏 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅))))
7 fveq2 6892 . . . . 5 (π‘Ž = suc 𝑏 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜suc 𝑏))
87sseq1d 4014 . . . 4 (π‘Ž = suc 𝑏 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅)))
98imbi2d 341 . . 3 (π‘Ž = suc 𝑏 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
10 fveq2 6892 . . . . 5 (π‘Ž = 𝐴 β†’ (πΉβ€˜π‘Ž) = (πΉβ€˜π΄))
1110sseq1d 4014 . . . 4 (π‘Ž = 𝐴 β†’ ((πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅) ↔ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅)))
1211imbi2d 341 . . 3 (π‘Ž = 𝐴 β†’ ((πœ‘ β†’ (πΉβ€˜π‘Ž) βŠ† (πΉβ€˜π΅)) ↔ (πœ‘ β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))))
13 ssid 4005 . . . 4 (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)
14132a1i 12 . . 3 (𝐡 ∈ Ο‰ β†’ (πœ‘ β†’ (πΉβ€˜π΅) βŠ† (πΉβ€˜π΅)))
15 isf32lem.b . . . . . . 7 (πœ‘ β†’ βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯))
16 suceq 6431 . . . . . . . . . 10 (π‘₯ = 𝑏 β†’ suc π‘₯ = suc 𝑏)
1716fveq2d 6896 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (πΉβ€˜suc π‘₯) = (πΉβ€˜suc 𝑏))
18 fveq2 6892 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘))
1917, 18sseq12d 4016 . . . . . . . 8 (π‘₯ = 𝑏 β†’ ((πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯) ↔ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2019rspcv 3609 . . . . . . 7 (𝑏 ∈ Ο‰ β†’ (βˆ€π‘₯ ∈ Ο‰ (πΉβ€˜suc π‘₯) βŠ† (πΉβ€˜π‘₯) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2115, 20syl5 34 . . . . . 6 (𝑏 ∈ Ο‰ β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
2221ad2antrr 725 . . . . 5 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘)))
23 sstr2 3990 . . . . 5 ((πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π‘) β†’ ((πΉβ€˜π‘) βŠ† (πΉβ€˜π΅) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅)))
2422, 23syl6 35 . . . 4 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ (πœ‘ β†’ ((πΉβ€˜π‘) βŠ† (πΉβ€˜π΅) β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
2524a2d 29 . . 3 (((𝑏 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝑏) β†’ ((πœ‘ β†’ (πΉβ€˜π‘) βŠ† (πΉβ€˜π΅)) β†’ (πœ‘ β†’ (πΉβ€˜suc 𝑏) βŠ† (πΉβ€˜π΅))))
263, 6, 9, 12, 14, 25findsg 7890 . 2 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ 𝐡 βŠ† 𝐴) β†’ (πœ‘ β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅)))
2726impr 456 1 (((𝐴 ∈ Ο‰ ∧ 𝐡 ∈ Ο‰) ∧ (𝐡 βŠ† 𝐴 ∧ πœ‘)) β†’ (πΉβ€˜π΄) βŠ† (πΉβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  π’« cpw 4603  βˆ© cint 4951  ran crn 5678  suc csuc 6367  βŸΆwf 6540  β€˜cfv 6544  Ο‰com 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fv 6552  df-om 7856
This theorem is referenced by:  isf32lem2  10349  isf32lem3  10350
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