Theorem isorel 7327

Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)

Assertion

Ref	Expression
isorel	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))

Proof of Theorem isorel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-isom 6553	. . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
2	1	simprbi 495	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
3		breq1 5152	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝐶𝑅𝑦))
4		fveq2 6892	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐻‘𝑥) = (𝐻‘𝐶))
5	4	breq1d 5159	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)))
6	3, 5	bibi12d 344	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦))))
7		breq2 5153	. . . 4 ⊢ (𝑦 = 𝐷 → (𝐶𝑅𝑦 ↔ 𝐶𝑅𝐷))
8		fveq2 6892	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐻‘𝑦) = (𝐻‘𝐷))
9	8	breq2d 5161	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐻‘𝐶)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
10	7, 9	bibi12d 344	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))))
11	6, 10	rspc2v 3623	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))))
12	2, 11	mpan9 505	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   class class class wbr 5149  –1-1-onto→wf1o 6543  ‘cfv 6544   Isom wiso 6545

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-isom 6553

This theorem is referenced by:  soisores  7328  isomin  7338  isoini  7339  isopolem  7346  isosolem  7348  weniso  7355  smoiso  8366  supisolem  9472  ordiso2  9514  cantnflt  9671  cantnfp1lem3  9679  cantnflem1b  9685  cantnflem1  9688  wemapwe  9696  cnfcomlem  9698  cnfcom  9699  cnfcom3lem  9702  fpwwe2lem5  10634  fpwwe2lem6  10635  fpwwe2lem8  10637  leisorel  14427  seqcoll  14431  seqcoll2  14432  isercoll  15620  ordthmeolem  23527  iccpnfhmeo  24692  xrhmeo  24693  dvcnvrelem1  25768  dvcvx  25771  isoun  32188  erdszelem8  34485  erdsze2lem2  34491  cantnfresb  42378  fourierdlem20  45143  fourierdlem46  45168  fourierdlem50  45172  fourierdlem63  45185  fourierdlem64  45186  fourierdlem65  45187  fourierdlem76  45198  fourierdlem79  45201  fourierdlem102  45224  fourierdlem103  45225  fourierdlem104  45226  fourierdlem114  45236