Theorem isorel 5980

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 5360	. . 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 275	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
3		breq1 4111	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝐶𝑅𝑦))
4		fveq2 5669	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐻‘𝑥) = (𝐻‘𝐶))
5	4	breq1d 4118	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)))
6	3, 5	bibi12d 235	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦))))
7		breq2 4112	. . . 4 ⊢ (𝑦 = 𝐷 → (𝐶𝑅𝑦 ↔ 𝐶𝑅𝐷))
8		fveq2 5669	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐻‘𝑦) = (𝐻‘𝐷))
9	8	breq2d 4120	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐻‘𝐶)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
10	7, 9	bibi12d 235	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))))
11	6, 10	rspc2v 2933	. 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 281	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520   class class class wbr 4108  –1-1-onto→wf1o 5350  ‘cfv 5351   Isom wiso 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-isom 5360

This theorem is referenced by:  isoresbr  5981  isoini  5990  isopolem  5994  isosolem  5996  smoiso  6532  isotilem  7296  supisolem  7298  ordiso2  7325  leisorel  11202  zfz1isolemiso  11204  seq3coll  11207